Before we move on to momentum,
we actually have a few last notes about the nature of forces
and conservative versus nonconservative forces.
If we have an apple, another example object,
moving up and down under the influence of gravity,
it turns out that gravity can only convert the energy of this object
between kinetic and potential.
So it's not taking the energy away or putting it anywhere else,
it's just converting it between kinetic and potential
without actually changing, as we saw, the overall energy of the object,
in this case the apple, through any entire path.
This means that throughout the entire path
and at each point in this path as the apple moves up and down,
the total energy is constant. It's not changing,
which is why we were able to use the conservation of energy
in the way that we did so far.
The important thing about this is that the entire work done
on this object as it moves up and down,
again by calculating force times distance in the direction
that your force is acting, of the distance of course
just being vertically up and vertically down
in the direction of the force which is vertically down.
The entire work done on this object
only depends on the path, or sorry,
the final and initial locations and not at all on the path.
A different way to say this is that
if I know my apple started at one position,
and then I know it followed some complicated path
to get to some final position,
I can tell you how much work gravity did
simply by the initial and final locations
without needing to know anything about the particular path
that it took getting there.
This is simply because the work done by gravity
as the object goes up and the work done by gravity as the object goes down,
will cancel regardless of which path it took to do that.
We can illustrate this a little better
by one more time reiterating the work done
by gravity is the same for any path between these two points,
because we only care about the initial and final location.
So, for example, I could have gone straight across horizontally.
I could have lifted the object and then I could have gone back down,
or I could have gone through some more complicated path.
But the work done by gravity in getting from point A to point B,
the two darker images here,
is the same for any of these paths that I've shown here.
The work in this particular example for paths 1, 2, and 3,
because the object ends in the same height,
the same location as it was initially, must be zero.
The reason for this is that we haven't traded off any potential energy,
we haven't gone down or up,
and so we also can't have changed our kinetic energy,
and so if we didn't change our kinetic energy,
the entire work done throughout the entirety of the path must be zero.
And this is the definition of a conservative force.
The work done by a conservative force
does not depend at all on the path that you took from getting
from one point to another,
it only depends on the initial and the final location of your object.
What we can do is see why the work done by the gravity
in these three paths is always going to be the same.
So in this first path,
we know that the object went vertically up and vertically down.
The fact that it went sideways, as it was doing this,
won't contribute at all to the work because remember
the work only cares about force and distance in the same direction.
So that cosine of theta term that we had in our expression for work
will be zero for any of the horizontal motion.
So, in other words,
the work done by gravity in just going up and then just going down
will only be up and down and will only be in the same magnitude up and down.
And this will always cancel to zero
if your object started in the same location that it ended.
Similarly, for the second path, we have a motion that is purely sideways,
and so if the distance is directly to the right while the force is downwards,
our work equation with cosine of theta in it
will tell us that no work was done because the force and the direction
of the displacement were perpendicular.
The work done by gravity depends, as I just said,
only on the initial and final locations regardless of the path
and this energy is going to be conserved
because the gravitational force, as we said,
only depends on the final and initial location,
only trading energy between potential and kinetic
without taking the total energy away or adding anything to the total energy.